Related question: this topic is examined
Diagonalization
A = PDP⁻¹ and fast powers Aᵏ = PDᵏP⁻¹, the Diagonalization Theorem (n independent eigenvectors), the step-by-step procedure, and when distinct eigenvalues guarantee diagonalizability.
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Notes
A is diagonalizable ⇔ it has n linearly independent eigenvectors. Then A = PDP⁻¹: P’s columns are eigenvectors, D’s diagonal the matching eigenvalues. Aᵏ = PDᵏP⁻¹.
Memory trick n distinct eigenvalues ⇒ diagonalizable (sufficient, not necessary).
Exam angle
Related question: this topic is examined
Related question: this topic is examined
A 3×3 matrix with 3 distinct eigenvalues is diagonalizable.
Every diagonalizable matrix is invertible.
- 0 can be an eigenvalue.
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Notes
Steps: eigenvalues → basis of each eigenspace → if the total is n vectors, build P and D (same order).
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